Problem: Express $z_1=6+2\sqrt{3}i$ in polar form. Express your answer in exact terms, using degrees, where your angle is between $0^\circ$ and $360^\circ$, inclusive. $z_1=$
The Strategy A complex number in rectangular form, $z={a}+{b}i$, can be written in polar form as $z={r}[\cos{\theta}+i\sin{\theta}]$, where ${r}$ is the absolute value, or modulus, and ${\theta}$ is the angle, or argument. Therefore, ${r}$ and ${\theta}$ can be found using the following formulas: ${r}=\sqrt{{a}^2+{b}^2}$ $\tan{\theta}=\dfrac{{b}}{{a}}$ [How did we get these equations?] Similarly, a complex number in polar form, $z={r}[\cos{\theta}+i\sin{\theta}]$, can be written in rectangular form as $z={a}+{b}i$, using the following formulas: ${a}={r}\cos{\theta}$ ${b}={r}\sin{\theta}$ [How did we get these equations?] Finding $r$ For $z_1={6}+{2\sqrt{3}}i$ : ${a} = {6}$ ${b} = {2\sqrt{3}}$ Therefore, we can find ${r}$ as follows. $\begin{aligned}{r}&=\sqrt{{a}^2+{b}^2} \\\\&=\sqrt{{6}^2+({2\sqrt{3}})^2} \\\\&=\sqrt{36+12} \\\\&={\sqrt{48}} \\\\&={4\sqrt{3}}\end{aligned}$ Finding $\theta$ Using the formula, we have: $\begin{aligned}{\theta}&=\arctan\left(\dfrac{{b}}{{a}}\right) \\\\&=\arctan\left(\dfrac{{2\sqrt{3}}}{{6}}\right) \\\\&={30^\circ}\end{aligned}$ Since ${a}$ and ${b}$ are both positive, ${\theta}$ must lie in Quadrant $\text{I}$. Therefore its angle must be between $0^\circ$ and $90^\circ$, which agrees with our number above. Summary $z_1={4\sqrt{3}}[\cos{30^\circ}+i\sin{30^\circ}]$